Properties

Label 2.12513.4t3.a.a
Dimension $2$
Group $D_{4}$
Conductor $12513$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(12513\)\(\medspace = 3 \cdot 43 \cdot 97 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.4.1213761.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: even
Determinant: 1.12513.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{97}, \sqrt{129})\)

Defining polynomial

$f(x)$$=$ \( x^{4} - 2x^{3} - 19x^{2} + 20x + 3 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 4 + 30\cdot 31 + 6\cdot 31^{2} + 9\cdot 31^{3} +O(31^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 12 + 19\cdot 31 + 26\cdot 31^{2} + 2\cdot 31^{3} + 21\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 20 + 11\cdot 31 + 4\cdot 31^{2} + 28\cdot 31^{3} + 9\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 28 + 24\cdot 31^{2} + 21\cdot 31^{3} + 30\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

Cycle notation
$(1,2)(3,4)$
$(2,3)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,4)(2,3)$$-2$
$2$$2$$(1,2)(3,4)$$0$
$2$$2$$(1,4)$$0$
$2$$4$$(1,3,4,2)$$0$

The blue line marks the conjugacy class containing complex conjugation.